To solve the problem involving a smooth wire bent into a vertical circle with a sliding block, we need to analyze the forces acting on the block and apply some principles of circular motion. Let's break it down step by step.
Understanding the System
We have a vertical circular wire of radius \( a \), and a block \( B \) that can slide along this wire. The entire setup is rotating about the diameter \( AB \) with an angular speed \( \Omega \). The block is at rest relative to the circular ring, which means it is not moving up or down the wire despite the rotation.
Forces Acting on the Block
When the block is at rest in the rotating frame, the forces acting on it include:
- The gravitational force acting downward, \( mg \), where \( m \) is the mass of the block and \( g \) is the acceleration due to gravity.
- The normal force exerted by the wire on the block, which acts perpendicular to the surface of the wire.
Applying Circular Motion Principles
Since the block is in a circular motion, we can analyze the centripetal force required to keep it moving in a circle of radius \( a \). The centripetal force \( F_c \) is given by:
F_c = m \cdot a_c = m \cdot \frac{v^2}{r}
Here, \( v \) is the tangential speed of the block, and \( r \) is the radius of the circular path, which is \( a \) in this case.
Relating Angular Speed to Linear Speed
The relationship between the linear speed \( v \) and the angular speed \( \Omega \) is:
v = \Omega \cdot r = \Omega \cdot a
Substituting this into the centripetal force equation gives us:
F_c = m \cdot \frac{(\Omega a)^2}{a} = m \Omega^2 a
Equilibrium of Forces
In the vertical position, the forces acting on the block must balance. The normal force \( N \) and the weight \( mg \) must satisfy the following equation:
N - mg = -m \Omega^2 a
Here, the negative sign indicates that the centripetal force is directed towards the center of the circular path. Rearranging gives:
N = mg - m \Omega^2 a
Condition for the Block to be at Rest
For the block to remain at rest with respect to the circular ring, the normal force \( N \) must be non-negative:
mg - m \Omega^2 a \geq 0
This leads us to the condition:
g \geq \Omega^2 a
Final Expression for Angular Speed
From the inequality derived, we can express the angular speed \( \Omega \) in terms of the gravitational acceleration \( g \) and the radius \( a \):
\(\Omega^2 \leq \frac{g}{a}\)
This means that the maximum angular speed at which the block can remain at rest on the wire is given by:
\(\Omega^2 = \frac{g}{a}\)
In summary, the angular speed \( \Omega \) must satisfy the condition derived from the balance of forces acting on the block in the rotating frame. This ensures that the block remains stationary relative to the wire while the system rotates.
